EP2241052B1 - Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln - Google Patents

Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln Download PDF

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Publication number
EP2241052B1
EP2241052B1 EP08863527.1A EP08863527A EP2241052B1 EP 2241052 B1 EP2241052 B1 EP 2241052B1 EP 08863527 A EP08863527 A EP 08863527A EP 2241052 B1 EP2241052 B1 EP 2241052B1
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Prior art keywords
key
traceable
keys
private key
private keys
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English (en)
French (fr)
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EP2241052A1 (de
Inventor
Alexandre Karlov
Pascal Junod
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Nagravision SARL
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Nagravision SA
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/304—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy based on error correction codes, e.g. McEliece
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/3066—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves
    • H04L9/3073—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves involving pairings, e.g. identity based encryption [IBE], bilinear mappings or bilinear pairings, e.g. Weil or Tate pairing
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/3093—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving Lattices or polynomial equations, e.g. NTRU scheme
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
    • H04L2209/34—Encoding or coding, e.g. Huffman coding or error correction
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
    • H04L2209/60—Digital content management, e.g. content distribution
    • H04L2209/601—Broadcast encryption
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
    • H04L2209/60—Digital content management, e.g. content distribution
    • H04L2209/606—Traitor tracing

Definitions

  • the aim of the present invention is to propose a very fast alternative mechanism to the traitor tracing algorithm introduced by Boneh and Franklin to trace private keys in a public-key cryptosystem.
  • a management center would like to broadcast some data to l receivers, where only authorized users (typically, those who have paid a fee) can have access to the data.
  • a possible solution widely deployed in commercial Pay-TV systems or in secured media distribution systems, for instance, consists in encrypting the data using a symmetric key and to securely transmit to each authorized receiver this key which will be stored in a tamper-proof piece of hardware, like a smartcard.
  • tamper-resistant hardware is very difficult and/or costly to design, since it is vulnerable to a wide variety of attacks. Therefore, a malicious user (hereafter called a traitor ) can try to retrieve the decryption key from his receiver and distribute it (sell or give away) to unauthorized users ( the pirates ) .
  • a malicious user hereafter called a traitor
  • a traitor can try to retrieve the decryption key from his receiver and distribute it (sell or give away) to unauthorized users (the pirates ) .
  • a dishonest user will try to mix several legitimate keys in order to build a new one and embed it in a pirate receiver device.
  • traitor tracing The problem of identifying which receivers were compromised and/or which secret keys were leaked is called traitor tracing.
  • traitor tracing Two modes of traitor tracing are considered: in the black-box mode, the tracing algorithm sends crafty ciphertexts to the rogue receiver and aims at determining which keys it uses while observing its behavior; in the non-black-box model, we assume that the keys (or their combination) can be extracted from the pirate receiver and are known to the tracing algorithm.
  • Fiat and Naor [1] introduced the concept of broadcast encryption.
  • the broadcasting center can dynamically specify a privileged subset of authorized users that can decrypt selected ciphertexts (like high-value content, for instance).
  • Chor, Fiat, and Naor [2] introduced the concept of traitor-tracing to thwart the problem of decryption keys piracy in broadcast encryption schemes; their scheme is k-collusion resistant (or k-resilient) in the sense that at least one traitor is identified with very high probability if there are at most k of them.
  • Boneh and Franklin [6] proposed a new public-key traitor-tracing scheme based on error-correcting codes, more precisely on Reed-Solomon codes.
  • the Boneh-Franklin non-black-box traitor tracing scheme is k-collusion resistant and deterministic in the sense that all of the traitors are identified with probability 1 as long as at most k of them collude to derive new pirate keys.
  • a linear code C over GF ( q ) l vector space is a subspace of GF ( q ) l .
  • C has a dimension 0 ⁇ 2 k ⁇ l and contains q 2 k codewords.
  • the minimal distance d of C is equal to the minimum Hamming weight of all non-zero codewords.
  • An l ⁇ 2 k matrix G over GF ( q ) is called a generator matrix if its columns form a linearly independent basis for C.
  • c (1,1,...,1), one simply speaks of Reed-Solomon codes.
  • the aim of the present application is a new non-black-box traitor tracing mechanism for Boneh-Franklin and related schemes. Due to its algorithmic complexity, the original traitor tracing procedure, as it was described by Boneh and Franklin, is limited to the systems of about 10'000 users in a practical scenario on a common desktop computer available in 2007.
  • the benefit of the present application is the possibility to perform key tracing in systems up to several billions of users in a fraction of time of the original tracing mechanism on the same computer configuration.
  • the identities of the traitors are revealed by subtracting ⁇ - ⁇ c ; the base point ⁇ ⁇ then reveals the index i of the key used in the coalition. In other words, it is the i -th non-zero element of the resulting vector.
  • the running time of the above algorithm is in O (l 3 ), i.e. it depends on the cube of the number l of users in the system. Such running time is prohibitive in practice as soon as l becomes larger than 10'000. Even though the complexity of the optimized version of the tracing procedure using the so-called Berlekamp-Welch described in [9] is in O (l 2 ) , the tracing capacity in terms of number of users in the system still remains heavily limited.
  • the goal of the algorithm is to find the subset T .
  • the new fast traitor tracing mechanism that is the subject of the present invention will take 2k elements as input and will output the identities of the traitors as long as there are at most k traitor keys in the representation d .
  • the new proposed algorithm consists of 2 steps:
  • the actual identity of the traitors is given through a identity-base-point relations table which explicitly maps the identity of a user to the corresponding base point of a given private key and vice-versa.
  • the first advantage of the proposed method is that the first step of the algorithm is independent of the number of users l in the system and only depends on the coalition size k which is negligible compared to the number of users in the system l in terms of complexity.
  • the second step is linear in l , but involves only light computations.
  • the tracing procedure can be performed considerably faster than in the original Boneh-Franklin proposal.
  • the overall complexity of the new algorithm is in O (l), i.e. it is linear in l .
  • the present invention can be applied, but is not limited, to the Pay-TV and other digital media distribution systems, such as CDs and DVDs.
  • the data is to be encrypted using the system's public key and each user is given a distinct private traceable key by means of incorporation of the abovementioned private key parts in tamper-proof memory of an electronic device, also referred as "receiver" (Set-Top Box, mobile terminal or an optical disc player).
  • an unlawful user also referred as "traitor”
  • an unlawful user would try to extract the private key information by means of reverse engineering, that is taking the receiver apart and discovering the key material through analysis of receiver's function and operations.
  • the traitor might then build his own receiver or player which utilizes a combination of the retrieved private keys in order to decrypt the protected media. He may sell it or he may put it on the Internet if the receiver can be simulated by a software program executing on a personal computer which will then result in the propagation of piracy.
  • the media distribution authority after locating and recovering the abovementioned pirate device and extracting the pirate key, will run the fast traitor tracing mechanism which is subject to the current disclosure.
  • the Berlekamp-Massey algorithm represented by the block BERMA-ALG will be executed on the traceable part of the pirate key PRT-KEY which consists of 2k elements and it will output the error locator polynomial ERR-LCR consisting of k coefficients.
  • the decision diamond KORL-DST-RTS it is determined whether the error locator polynomial has k or fewer roots, which corresponds to the pirate private key being formed by k or less original private keys according to (1).
  • the logic flows from decision diamond KORL-DST-RTS to output the identities of the traitors TR-ID by searching the modular inverse of the roots of the error locator polynomial against the identity-base-points relations table.
  • the main advantage of the current invention is its speed, comparing to the previously proposed art, meaning that a system implementing the fast traitor tracing can be deployed in a scenario with several hundreds of millions of receivers, which seems deeply appropriate to the digital media distribution systems such as those using optical discs (DVDs and CDs) and Pay-TV systems, whereas previously known art [6] could only be used in a scenario with 10'000 of users which may be seen as a severe limiting factor.
  • the fast traitor tracing uses the non-black-box deterministic tracing approach. It means that first of all, the key material is to be extracted from the receiver by media distribution authority by means of reverse engineering. The advantage of this approach is the deterministic nature of the fast traitor tracing mechanism, specifically until there is a combination of k or less rogue keys in the pirate receiver, the identities of the traitors will be discovered with probability 1. On the contrary, the method proposed in [4], under some circumstances, will not be able to output the exact identities of the traitors.

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  • Engineering & Computer Science (AREA)
  • Theoretical Computer Science (AREA)
  • Signal Processing (AREA)
  • Computing Systems (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • Computer Security & Cryptography (AREA)
  • Algebra (AREA)
  • General Physics & Mathematics (AREA)
  • Pure & Applied Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Physics (AREA)
  • Mathematical Optimization (AREA)
  • Physics & Mathematics (AREA)
  • Error Detection And Correction (AREA)
  • Storage Device Security (AREA)
  • Detection And Correction Of Errors (AREA)

Claims (3)

  1. Verfahren zur Rückverfolgung rückverfolgbarer Teile d = (d1,...,d2k)T von privaten Originalschlüsseln in einem Kryptosystem mit öffentlichem Schlüssel, bestehend aus einem öffentlichen Schlüssel und ℓ entsprechenden privaten Schlüsseln, wobei ein privater Schlüssel aus mindestens einer rückverfolgbaren Anordnung von 2k Elementen besteht, die ein Syndrom eines verallgemeinerten Reed-Solomon-Codes bilden, mit Parametern (ℓ,ℓ-2k) definiert durch die Basispunkte π = (π1,...,πl)T und einen Skalierungsvektor c = (c1,...,cl)T, in dem k eine maximale Koalitionsgröße ist, T eine Teilmenge von kompromittierten Schlüsseln, ℓ die Zahl der Benutzer und q eine große Primzahl, mit folgenden Phasen:
    - Erhalt des rückverfolgbaren Teils d = (d1,...,d2k)T eines privaten Rogue-Schlüssels, wobei die Anordnung von 2k Elementen eine lineare Kombination von Anordnungen von 2k Elementen ist, die mindestens zwei privaten Originalschlüsseln angehören, so dass die Summe der Koeffizienten der linearen Kombination gleich 1 mod q ist,
    gekennzeichnet dadurch, dass es des weiteren folgende Phasen umfasst:
    - Anwendung eines Berlekamp-Massey-Algorithmus auf den rückverfolgbaren Teil d = (d1,...,d2k)T des privaten Rogue-Schlüssels, um die k Koeffizienten eines Fehlerstellenpolynoms zu erhalten,
    - Anwendung von Chiens Suchalgorithmus auf das Fehlerstellenpolynom, um Wurzeln des Fehlerstellenpolynoms zu erhalten,
    - Bestimmung der Basispunkte des rückverfolgbaren Teils der originalen privaten Schlüssel durch Berechnung der arithmetischen Umkehrfunktion jeder Wurzel, wobei es diese Basispunkte gestatten, die Identität des privaten Schlüssels eindeutig zu bestimmen.
  2. Verfahren nach Anspruch 1, wobei das verallgemeinerte Reed-Solomon-Codewort über ein endliches Feld von mindesten 2127 Elementen definiert ist.
  3. Verfahren nach Anspruch 1 und 2, wobei jeder private Originalschlüssel einer gegebenen Identität zugeteilt wird, wobei das Verfahren des weiteren folgende Phasen umfasst:
    - Speicherung, für jeden privaten Schlüssel, des Basispunkts πi und der entsprechenden Identität i,
    - Wiederauffinden der Identitäten der privaten Schlüssel auf der Grundlage des wiedergewonnenen Basispunkts und der entsprechenden gespeicherten Identitäten.
EP08863527.1A 2007-12-21 2008-12-18 Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln Active EP2241052B1 (de)

Priority Applications (1)

Application Number Priority Date Filing Date Title
EP08863527.1A EP2241052B1 (de) 2007-12-21 2008-12-18 Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln

Applications Claiming Priority (3)

Application Number Priority Date Filing Date Title
EP07150363A EP2073431A1 (de) 2007-12-21 2007-12-21 Verfahren zur Rückverfolgung rückverfolgbarer Teile privater Originalschlüssel in einem Verschlüsselungssystem mit öffentlichen Schlüsseln
PCT/EP2008/067827 WO2009080683A1 (en) 2007-12-21 2008-12-18 Method to trace traceable parts of original private keys in a public-key cryptosystem
EP08863527.1A EP2241052B1 (de) 2007-12-21 2008-12-18 Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln

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EP2241052A1 EP2241052A1 (de) 2010-10-20
EP2241052B1 true EP2241052B1 (de) 2013-05-15

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EP07150363A Withdrawn EP2073431A1 (de) 2007-12-21 2007-12-21 Verfahren zur Rückverfolgung rückverfolgbarer Teile privater Originalschlüssel in einem Verschlüsselungssystem mit öffentlichen Schlüsseln
EP08863527.1A Active EP2241052B1 (de) 2007-12-21 2008-12-18 Verfahren zur rückverfolgung rückverfolgbarer teile privater originalschlüssel in einem verschlüsselungssystem mit öffentlichen schlüsseln

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EP07150363A Withdrawn EP2073431A1 (de) 2007-12-21 2007-12-21 Verfahren zur Rückverfolgung rückverfolgbarer Teile privater Originalschlüssel in einem Verschlüsselungssystem mit öffentlichen Schlüsseln

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EP (2) EP2073431A1 (de)
KR (1) KR101439587B1 (de)
CN (1) CN101897149B (de)
WO (1) WO2009080683A1 (de)

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CN112769573B (zh) * 2020-12-26 2022-08-05 西安电子科技大学 一种基于grs码的数字签名方法、验签方法及其装置

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US4633470A (en) 1983-09-27 1986-12-30 Cyclotomics, Inc. Error correction for algebraic block codes
US7039803B2 (en) 2001-01-26 2006-05-02 International Business Machines Corporation Method for broadcast encryption and key revocation of stateless receivers
US7010125B2 (en) 2001-01-26 2006-03-07 Interntional Business Machines Corporation Method for tracing traitor receivers in a broadcast encryption system
FR2856539A1 (fr) * 2003-06-17 2004-12-24 France Telecom Procede et systeme tracables de chiffrement et/ou de dechiffrement d'informations, et supports d'enregistrement pour la mise en oeuvre du procede

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KR20100093091A (ko) 2010-08-24
KR101439587B1 (ko) 2014-10-30
CN101897149A (zh) 2010-11-24
EP2073431A1 (de) 2009-06-24
WO2009080683A1 (en) 2009-07-02
EP2241052A1 (de) 2010-10-20
CN101897149B (zh) 2013-01-02

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